We will show that the signatures of the Lefschetz fibrations corresponding to the left hand side and right hand side of this equation are \(-7\)  and \(-1\)  respectively. Our computation is consistent with a result of Endo and Nagami [1]} where it was proven that the difference of the signatures of these two Lefschetz fibrations is \(-6\) . To compute \(\sigma _{\delta _1\delta _2}\)  and \(\sigma _{(C_3C_2C_1)^{4}},\)  attach a 1-handle connecting the two boundary components and then cap off the boundary. By above discussion, the signature is unchanged. Note that \(\delta _1=\delta _2=\delta \) 
in \({\rm H_1}(\widehat{\Sigma };{\mathbb {R}})\)  where \(\widehat{\Sigma }\)  is a surface below.
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